DeMoivre’s quintic and a theorem of Galois

Explicit formulae for the five roots of DeMoivre's quintic polynomial are given in terms of any two of the roots. If f ( x) is an irreducible polynomial of prime degree over the rational field Q, a classical theorem of Galois asserts that f ( x) is solvable by radicals if and only if all the roots of f ( x) can be expressed a s rational functions of any two of them, see for example [2, p. 2541. I t is known that DeMoivre's quintic polynomial is solvable by radicals, see for example Borger [I]. In this paper we give explicit formulae for the roots of f ( x) in terms of any two of them. We do not need to assume that f ( x) is irreducible only that it has nonzero discriminant, that is, d = t j5(4a5-b2)2 t O. (2) We remark that if d = 0 then 4a5 = b2 so that a = u2 and b = 2u5 for some u E Q and the roots of f ( x) ar